How can this code be extended to include a nice aesthetically looking main menu for this newton's method program with the equation being displayed to the user in a nice font upon running the program, showing the option to either adjust the sliders showing the graph, or run a simulation until no more iterations can be found highlighting each iteration in a graph using a legend or enter the values manually, the slider option should be fixed and working fully as it currently isn't working at all. Here is the code, the updated code should be shown in its entirety

import matplotlib.pyplot as plt

from ipywidgets import interact, FloatSlider, Button, HBox, VBox, Output

import numpy as np

# Define the function based on the equation: f$($x$)=$ ax$^$b $-$ e$^($cx$)*$ sin$($wx $+$ v$)$

def f$($x$,$ a$,$ b$,$ c$,$ w$,$ v$)$:

$$ return a $*$ x$**$b $-$ np$.$exp$($c $*$ x$)*$ np$.$sin$($w $*$ x $+$ v$)$

# Define the derivative of the function with respect to x

def f$\_$prime$($x$,$ a$,$ b$,$ c$,$ w$,$ v$)$:

$$ term$1=$ b $*$ a $*$ x$**($b $-1)$ # Derivative of ax$^$b

$$ term$2=$ np$.$exp$($c $*$ x$)*($c $*$ np$.$sin$($w $*$ x $+$ v$)+$ w $*$ np$.$cos$($w $*$ x $+$ v$))$

$$ return term$1-$ term$2$

# Function implementing Newton's method to find a root starting from an initial guess x$0$

def newtons$\_$method$($a$,$ b$,$ c$,$ w$,$ v$,$ x$0,$ tolerance$=1$e$-7,$ max$\_$iterations$=100)$:

$$ x $=$ x$0$

$$ for i in range$($max$\_$iterations$)$:

$$ fx $=$ f$($x$,$ a$,$ b$,$ c$,$ w$,$ v$)$

$$ fpx $=$ f$\_$prime$($x$,$ a$,$ b$,$ c$,$ w$,$ v$)$

$$ if fpx $==0$:

$$ print$("$Zero derivative. No solution found."$)$

$$ return None

$$ x$\_$new $=$ x $-$ fx $/$ fpx

$$ if abs$($x$\_$new $-$ x$)<$ tolerance:

$$ return x$\_$new

$$ x $=$ x$\_$new

$$ print$("$Exceeded maximum iterations. No solution found."$)$

$$ return None

# Function to find multiple roots by applying Newton's method from various initial guesses

def mult$\_$solns$($a$,$ b$,$ c$,$ w$,$ v$,$ initial$\_$guesses$)$:

$$ solutions $=[]$

$$ for x$0$ in initial$\_$guesses:

$$ solution $=$ newtons$\_$method$($a$,$ b$,$ c$,$ w$,$ v$,$ x$0)$

$$ if solution is not None:

$$ if all$($abs$($solution $-$ s$)>1$e$-5$ for s in solutions$)$:

$$ solutions.append$($solution$)$

$$ return solutions

# Function to plot the function and solutions on a graph

def plot$\_$function$($a$,$ b$,$ c$,$ w$,$ v$)$:

$$ initial$\_$guesses $=$ np$.$linspace$(-10,10,10)$# Define initial guesses

$$ solutions $=$ mult$\_$solns$($a$,$ b$,$ c$,$ w$,$ v$,$ initial$\_$guesses$)$# Find solutions

$$ # Plot the function

$$ t $=$ np$.$linspace$(-10,10,1000)$

$$ y $=$ f$($t$,$ a$,$ b$,$ c$,$ w$,$ v$)$

$$ plt$.$figure$($figsize$=(10,6))$

$$ plt$.$plot$($t$,$ y$,$ label$="$f$($x$)")$